Unusual topological properties
of the Nevanlinna class

 

Joel H. Shapiro and Allen L. Shields

 

American J. Math. 97 (1975), 915--936
 

Abstract: This paper considers the Nevanlinna class N of functions that are analytic in the open unit disc with bounded characteristic, and its subclass N+, the Smirnov class, both endowed with a natural complete metric in which N+ becomes a closed metric subspace of N. We prove that both N and the quotient space N/N+ are disconnected, and show that every finite-dimensional linear subspace of N/N+ has the discrete topology (but that, nevertheless, N/N+ is not itself discrete). Furthermore, we give an example of an infinite-dimensional linear subsapce of N/N+ that is discrete.

We generalize our results to other domains in the complex plane and to the polydisc and unit ball in C^n, and also consider corresponding problems for entire functions of finite order. We present a number of open questions. The most important one seeks to know if N+ is the component of the origin in N; this was answered later in the negative by James W. Roberts [The component of the origin in the Nevanlinna class, Illinois J. Math. 19 (1975), 553559] who used the Corona Theorem to prove that the component of the origin is the closed subspace of N consisting of functions whose "denominator singular measure" has no mass points.

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